Problem 67 - Elementary Algebra

Question

Simplify. $$ 5(2 x-3)+7 $$

Step-by-Step Solution

Answer

The simplified expression is \(10x - 8\).

Step 1: Apply Distribution

First, distribute the number outside the parentheses, which is 5, to each term within the parentheses. The expression is \(5(2x - 3)\). This means you calculate \(5 \times 2x\) and \(5 \times (-3)\). This results in the expression \(10x - 15\).

Step 2: Combine Like Terms

Now take the distributed expression \(10x - 15\) and add the number \(+7\) from the original expression. Combine \(-15 + 7\) to simplify it to \(-8\). Therefore, the resulting expression will be \(10x - 8\).

Key Concepts

Distribution Property

In mathematics, the distribution property is a crucial tool utilized in simplifying expressions. It involves distributing a factor equally across all terms inside a parenthesis. Consider the expression \(5(2x - 3)\). The number 5 outside the parenthesis needs to be multiplied by each term within, thus applying the distribution property.

It's done as follows:

• Multiply 5 by \(2x\), resulting in \(10x\).
• Multiply 5 by \(-3\), resulting in \(-15\).

The expression \(5(2x - 3)\) is now simplified to \(10x - 15\). Using the distribution property makes complex expressions easier to handle, especially when dealing with linear expressions. It allows us to break down and simplify each component systematically. Recognizing where to apply this property is key to developing more intuitive algebra problem-solving skills.

Combining Like Terms

After using the distribution property, the next step is to further simplify the expression by combining like terms. Let's look at the expression we derived: \(10x - 15 + 7\). Here, combining like terms means we group together and simplify parts of the expression that have identical variable parts or no variable at all.

In this example:

• The term \(10x\) stands alone, since there are no other \(x\) terms, it remains as it is.
• The constant terms \(-15\) and \(+7\) are the same type, so we can add them together to get \(-8\).

Therefore, after combining, the simplified expression becomes \(10x - 8\). Combining like terms is about identifying and merging terms that are similar, which streamlines the expression considerably.

Linear Expressions

Linear expressions form a foundational concept in algebra. They are algebraic expressions that do not involve exponents higher than 1. In simple terms, linear expressions have variables raised only to the first power, and they do not have products of variables. The exercise we simplified \(5(2x - 3) + 7\) ultimately results in \(10x - 8\), which is a classic linear expression.

Linear expressions are characterized by:

• Terms with variables that have no exponents or an exponent of 1, like \(x\).
• Being graphically represented as straight lines when plotted on a coordinate plane.
• Simplifying through operations such as distribution and combining like terms, just as we did in the exercise.

Understanding linear expressions helps in solving equations, understanding functions, and analyzing relationships between quantities. They lay the groundwork for more complex algebra topics explored later in mathematics.